Apply Kalman Filter in Financial Time Series

Transcription

1 Apply Kalman Filter in Financial Time Series Final Project for EE616 Signal Detection & Estimation Xingzhong Xu Department of Electrical & Computer Engineering Stevens Institute of Technology April 9, 01 Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, 01 1 / 3

2 Introduction Financial time series are well-known non-stationary. There s no perfect prediction model for such time series. A fundamental assumption is that the underlying series are driven by some hidden control or variables. A good approximate model should, demonstrates the hidden effects (state-space model) provide a good prediction performance (mean square error) computationally efficient (recursive filtering) In this project, I will use dynamic state-space system to model the financial time series, and then use Kalman filter to efficiently make prediction. Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, 01 / 3

3 Review Kalman Filter Under a Gaussian-Markov state model (u[n] N(0,Q)) s[n] = As[n 1]+Bu[n] and Bayesian linear observation model (w[n] N(0, C[n])) x[n] = H[n]s[n]+w[n] a Kalman filter is a recursive (prediction & correction only use present input x[n] and previous output ŝ[n 1 n 1], K[n] is Kalman Gain 1 ) ŝ[n n 1] = Aŝ[n 1 n 1] ŝ[n n] = ŝ[n n 1]+K[n](x[n] H[n]ŝ[n n 1]) MMSE estimator (M are minimum mean square error matrix). M[n n 1] = AM[n 1 n 1]A T +BQB 1 M[n n] = (I K[n]H[n])M[n n 1] 1 K[n] = M[n n 1]H T [n](c[n]+h[n]m[n n 1]H T [n]) 1 Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, 01 3 / 3

4 Basic Model In finance, compare to the assets price p, the rate of return r tend to behavior more stationary. We denote r as r[n] = log(p[n]) log(p[n 1]) Although the true value of r[n] is unknown, we could always observer it in noise market by, R[n] = r[n]+w[n] w[n] N(0,σ w ) In this project, I will analyzing two models with different assumptions as follows r is constant. r is mean reverting. Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, 01 4 / 3

5 Constant r We firstly assume the r is constant, then, r[n] = r[n 1]+u We further assume the observation and process noises are WSS (u N(0,σ u), w N(0,σ w)) and σ u σ w. Recall the Kalman filter discussion, we have ˆr[n n 1] = ˆr[n 1 n 1] M[n n 1] = M[n 1 n 1]+σu M[n n 1] K[n] = σw +M[n n 1] ˆr[n n] = ˆr[n n 1]+K[n](R[n] ˆr[n n 1]) M[n n] = (1 K[n])M[n n 1] Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, 01 5 / 3

6 Constant r - Parameter Estimation In the above model, we assume σ u, σ w and µ are constant parameters. Now we estimate them from real data. Recall the Gaussian Linear assumption and σ u σ w, R N(r,σ w I) r N(µ,σ ui) R N(µ,(σ w +σ u)i) R N(µ,σ w I) The MLE of γ = [ σ w µ ] T is given by, arg max γ L(γ R) Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, 01 6 / 3

7 Constant r - Parameter Estimation R N(µ,σ wi) logl(γ R) = logf(r,γ) logl(γ R) µ n (R[n] µ) = log exp( ) σw (πσ w ) N/ = N n log(πσ w)+ (R[n] µ) σ w n = (R[n] µ) (set to 0) σ w ˆµ = 1 R[n] N n = R[n] (MLE of µ) Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, 01 7 / 3

8 Constant r - Parameter Estimation logl(γ R) σw = N n σw (R[n] µ) σw 4 (set to 0) ˆσ w = 1 (R[n] µ) N n = 1 (R[n] N R[n]) (MLE of σw) n [ ] R[n] ˆγ = 1 N n (R[n] R[n]) Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, 01 8 / 3

9 Constant Model - Application Exxon Mobil Corporation(NYSE:XOM) historical daily price and return from to Use first 80% data to find the MLE of γ = [ % % ] T Daily Return (XOM) Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, 01 9 / 3

10 Constant Model - Application Use latest 0% data to recursively evaluate the ˆr[n n 1] Daily Return Predict Return Daily Price Predict Price Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, / 3

11 Mean-reverting Model Now we relax r s constant assumption. Let us assume E(r n ) = µ, and r is mean-reverting. r n r n 1 = α(µ r n 1 )+u Then the state space model will be given by r n = (1 α)r n 1 +αµ+u E(r) = µ var(r) = σ u α α The observation model will be given by R n = r n +w Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, / 3

12 Mean-reverting Model sample simulation µ = 0.1 σ u = alpha = 0.4 mu= alpha = 1.4 mu= Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, 01 1 / 3

13 Mean-reverting Model - Parameter Estimation In the above model, we assume the α, σ w, σ u and µ are unknown constant parameters. According to the linear Gaussian assumption, R n = r n +w = (1 α)r n 1 +αµ+u +w = (1 α)(r n 1 w)+αµ+u +w = (1 α)r n 1 +αµ+u +αw which shows R n is an autoregressive process AR(1). We would like to obtain the MLE of γ = [ α σ w σ u µ ] T, arg max γ L(γ R) Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, / 3

14 Mean-reverting Model - Conditional MLE R n R n 1 N((1 α)r n 1 +αµ,σ u +α σ w) f(r n R n 1,γ) 1 = π(σ u +α σw ) exp( (R n (1 α)r n 1 αµ) (σu +α σw) ) log(f(r n R n 1,γ)) = log(π(σ u +α σ w)) (R n (1 α)r n 1 αµ) (σ u +α σ w) Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, / 3

15 Mean-reverting Model - Marginal MLE Recall R is a stationary AR(1) process, we can assume R 1 as, E[R 1 ] = µ var[r 1 ] = σ u +α σw α α ( R 1 N µ, σ u +α σw ) α α ( π(σ f(r 1,γ) = u +α σw) ) 1/ exp( α α (R 1 µ) (α α ) ) (σu +α σw ) log(f(r 1,γ)) = 1 ( π(σ log u +α σw ) ) α α (R 1 µ) (α α ) (σu +α σw ) Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, / 3

16 Mean-reverting Model - Exact MLE logl(γ R) = logf(r 1,γ)+ f(r n,...,r 1 γ) = f(r 1,γ) n f(r n R n 1,γ) t= n logf(r t R t 1,γ) t= = 1 ( π(σ log u +α σw ) α α n ( log(π(σ u +α σw)) t= = log(α α ) ) (R 1 µ) (α α ) (σ u +α σ w) + (R t (1 α)r t 1 αµ) (σ u +α σ w) n log(π(σ u +α σ w)) (R 1 µ) (α α ) (σ u +α σ w) 1 n (σu +α σw ) (R t (1 α)r t 1 αµ) t= Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, / 3 )

17 Mean-reverting Model - Kalman filter Notice that the log-likelihood function log L(γ R) is a non-linear function, so there s no exact analytical solution for MLE ˆγ. here we use numerical method, arg max γ logl(γ R) We then use MLE γ to configure a Kalman filter. ˆr[n n 1] = (1 ˆα)ˆr[n 1 n 1]+ ˆαˆµ M[n n 1] = (1 ˆα) M[n 1 n 1]+ ˆσ u M[n n 1] K[n] = ˆσ w +M[n n 1] ˆr[n n] = ˆr[n n 1]+K[n](R[n] ˆr[n n 1]) M[n n] = (1 K[n])M[n n 1] Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, / 3

18 Mean-reverting Model sample simulation µ = 0.1 σ u = 0.1 σ w = 0.01 α = observation predict mu= Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, / 3

19 Mean-reverting Model - Application Exxon Mobil Corporation(NYSE:XOM) historical daily price and return from to Use first 80% data to find the MLE of γ = [ % 4.15% % ] T Daily Return (XOM) Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, / 3

20 Mean-reverting Model - Application Use latest 0% data to recursively evaluate the ˆr[n n 1] Daily Return Predict Return Daily Price Predict Price Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, 01 0 / 3

21 Constant versus Mean-reverting Model Mean-reverting model have better tracking error performance, especially when price change dramatically Daily Price Predict Price constant Predict Price mean reverting Square Error constant Square Error mean reverting Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, 01 1 / 3

22 Summary The financial time series in real applications are always non-stationary. So there s no perfect model can fit them well. I assume the daily return series are stationary, and thus using two state space model (constant and time-reverting) to model it separately. Both models parameters were estimated (analytically or numerically) through maximizing its likelihood function. Then based on the parameters, a configured Kalman filter is used to recursively predict and correct the underlying series. Not surprisingly, a more complicated mean-reverting model have better prediction performance than the constant one. Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, 01 / 3

23 Reference Franco JCG. Maximum likelihood estimation of mean reverting processes. Real Options Practice. Onward Inc. Welch, Greg, and Gary Bishop. An Introduction to the Kalman Filter. Ed. L Hansson, L Fahrig, & G Merriam. Design 7.1 (001) : Paresh Date. Kalman Filtering in Mathematical Finance. CARISMA, Brunel University. Eric Zivot. Estimation of ARMA Models. 005 Xingzhong Xu (Stevens) Apply Kalman Filter in Financial Time Series April 9, 01 3 / 3